286 SOME PROBLEMS CONCERNING THE MUTUAL INFLUENCE OF [390
A similar result may be arrived at from the value of ^ at an infinite distance, by use of the definite integral*
)sin0d0 = — ................... (32)
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As an example where the company of resonators extends to infinity, we may suppose that there is a row of them, equally spaced at distance K By (18)
-2ikR p-ZikR ]
+ + - ............. (33)
The series may be summed. If we write
7, ,,— 2ix 7,2 0-$ix
2 = *-* + ^- + --^- +..., .................. (34)
where h is real and less than unity, we have
l~he
and 2 = -log(l -&«-**), ........................ (35)
no constant of integration being required, since
2 = — hrl log (1 — h} when x = 0. If now we put h = 1,
2 = - log (1 - e~ix) = - log (2 sin jf) + #, (x - TT) + 2i ?ITT ....... (36)
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J
Thus T- = i-r5 \~ log f 2 sin —} + & (JcR - TT) + Zi
/C'Cl!' /Cx£ [ \ 4 /
If /^ = 2m7T, or ^ = rnX, where m is an integer, the logarithm becomes infinite and a tends to vanish -f*.
When R is very small, a is also very small, tending to
a = It -r- 2 log (kR) ............................ (38)
The longitudinal density of the now approximately linear source may be considered to be ajR, and this tends to vanish. The multiplication of resonators ultimately annuls the effect at a distance. It must be remembered that the tuning of each resonator is supposed to be as for itself alone.
In connexion with this we may consider for a moment the problem in two dimensions of a linear resonator parallel to the primary waves, which responds symmetrically. As before, we may take at the resonator
* Enc. Brit. 1. c. equation (43) ; Scientific Papers, Vol. in. p. 98.
t Phil. Mag. Vol. xiv. p. 60 (1907) ; Scientific Papers, Vol. v. p. 409.e first we consider the value of ty and its modulus at a great distance r from the resonators. It is evident that \/r is symmetrical with respect to the line R joining the resonators, and if 6 be the angle between r and R, r^ — rz = R cos 0. Thus